Ideal quotient

In abstract algebra, if I and J are ideals of a commutative ring R, their ideal quotient (I : J) is the set

(I:J)=\{r\in R\mid rJ\subseteq I\}

Then (I : J) is itself an ideal in R. The ideal quotient is viewed as a quotient because $KJ\subseteq I$ if and only if $K\subseteq (I:J)$ . The ideal quotient is useful for calculating primary decompositions. It also arises in the description of the set difference in algebraic geometry (see below).

(I : J) is sometimes referred to as a colon ideal because of the notation. In the context of fractional ideals, there is a related notion of the inverse of a fractional ideal.

Properties

The ideal quotient satisfies the following properties:

$(I:J)=\mathrm {Ann} _{R}((J+I)/I)$ as $R$ -modules, where $\mathrm {Ann} _{R}(M)$ denotes the annihilator of $M$ as an $R$ -module.
$J\subseteq I\Leftrightarrow (I:J)=R$ (in particular, $(I:I)=(R:I)=(I:0)=R$ )
$(I:R)=I$
$(I:(JK))=((I:J):K)$
$(I:(J+K))=(I:J)\cap (I:K)$
$((I\cap J):K)=(I:K)\cap (J:K)$
$(I:(r))={\frac {1}{r}}(I\cap (r))$ (as long as R is an integral domain)

Calculating the quotient

The above properties can be used to calculate the quotient of ideals in a polynomial ring given their generators. For example, if I = (f₁, f₂, f₃) and J = (g₁, g₂) are ideals in k[x₁, ..., x_n], then

I:J=(I:(g_{1}))\cap (I:(g_{2}))=\left({\frac {1}{g_{1}}}(I\cap (g_{1}))\right)\cap \left({\frac {1}{g_{2}}}(I\cap (g_{2}))\right)

Then elimination theory can be used to calculate the intersection of I with (g₁) and (g₂):

I\cap (g_{1})=tI+(1-t)(g_{1})\cap k[x_{1},\dots ,x_{n}],\quad I\cap (g_{2})=tI+(1-t)(g_{2})\cap k[x_{1},\dots ,x_{n}]

Calculate a Gröbner basis for $tI+(1-t)(g_{1})$ with respect to lexicographic order. Then the basis functions which have no t in them generate $I\cap (g_{1})$ .

Geometric interpretation

The ideal quotient corresponds to set difference in algebraic geometry.^[1] More precisely,

If W is an affine variety (not necessarily irreducible) and V is a subset of the affine space (not necessarily a variety), then

I(V):I(W)=I(V\setminus W)

where

I(\bullet )

denotes the taking of the ideal associated to a subset.

If I and J are ideals in k[x₁, ..., x_n], with k an algebraically closed field and I radical then

Z(I:J)=\mathrm {cl} (Z(I)\setminus Z(J))

where

\mathrm {cl} (\bullet )

denotes the Zariski closure, and

Z(\bullet )

denotes the taking of the variety defined by an ideal. If I is not radical, then the same property holds if we saturate the ideal J:

Z(I:J^{\infty })=\mathrm {cl} (Z(I)\setminus Z(J))

where

(I:J^{\infty })=\cup _{n\geq 1}(I:J^{n})

.

Examples

In $\mathbb {Z}$ we have $((6):(2))=(3)$ .
In algebraic number theory, the ideal quotient is useful while studying fractional ideals. This is because the inverse of any invertible fractional ideal $I$ of an integral domain $R$ is given by the ideal quotient $((1):I)=I^{-1}$ .
One geometric application of the ideal quotient is removing an irreducible component of an affine scheme. For example, let $I=(xyz),J=(xy)$ in $\mathbb {C} [x,y,z]$ be the ideals corresponding to the union of the x,y, and z-planes and x and y planes in $\mathbb {A} _{\mathbb {C} }^{3}$ . Then, the ideal quotient $(I:J)=(z)$ is the ideal of the z-plane in $\mathbb {A} _{\mathbb {C} }^{3}$ . This shows how the ideal quotient can be used to "delete" irreducible subschemes.
A useful scheme theoretic example is taking the ideal quotient of a reducible ideal. For example, the ideal quotient $((x^{4}y^{3}):(x^{2}y^{2}))=(x^{2}y)$ , showing that the ideal quotient of a subscheme of some non-reduced scheme, where both have the same reduced subscheme, kills off some of the non-reduced structure.
We can use the previous example to find the saturation of an ideal corresponding to a projective scheme. Given a homogeneous ideal $I\subset R[x_{0},\ldots ,x_{n}]$ the saturation of $I$ is defined as the ideal quotient $(I:{\mathfrak {m}}^{\infty })=\cup _{i\geq 1}(I:{\mathfrak {m}}^{i})$ where ${\mathfrak {m}}=(x_{0},\ldots ,x_{n})\subset R[x_{0},\ldots ,x_{n}]$ . It is a theorem that the set of saturated ideals of $R[x_{0},\ldots ,x_{n}]$ contained in ${\mathfrak {m}}$ is in bijection with the set of projective subschemes in $\mathbb {P} _{R}^{n}$ .^[2] This shows us that $(x^{4}+y^{4}+z^{4}){\mathfrak {m}}^{k}$ defines the same projective curve as $(x^{4}+y^{4}+z^{4})$ in $\mathbb {P} _{\mathbb {C} }^{2}$ .

References

^ David Cox; John Little; Donal O'Shea (1997). Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra. Springer. ISBN 0-387-94680-2., p.195
^ Greuel, Gert-Martin; Pfister, Gerhard (2008). A Singular Introduction to Commutative Algebra (2nd ed.). Springer-Verlag. p. 485. ISBN 9783642442544.

Viviana Ene, Jürgen Herzog: 'Gröbner Bases in Commutative Algebra', AMS Graduate Studies in Mathematics, Vol 130 (AMS 2012)

M.F.Atiyah, I.G.MacDonald: 'Introduction to Commutative Algebra', Addison-Wesley 1969.

[1] David Cox; John Little; Donal O'Shea (1997). Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra. Springer. ISBN 0-387-94680-2., p.195

[2] Greuel, Gert-Martin; Pfister, Gerhard (2008). A Singular Introduction to Commutative Algebra (2nd ed.). Springer-Verlag. p. 485. ISBN 9783642442544.

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Ideal quotient

Properties

Calculating the quotient

Geometric interpretation

Examples

References